Lattice-type antimetric filters



Dec. 30, 1958 Filed Nov. 30, 1953 J. R. V. OSWALD LATTICE-TYPE ANTIMETRIC FILTERS 4 Sheets-Sheet 1 mvENTo TAc uES RV. osuAl-b Dec. 30, 1958 QSWALD I I 2,866,951

LATTICE-TYPE ANTIMETRIC FILTERS Filed Nov. so, 1953 4 Sheets-Shet 2 INVEHTOK.

TAcous R.v. oswA D Dec. 30, 1958 v, osw D 2,866,951

LATTICE-TYPE ANTIMETRIC FILTERS Filed Nov. 30, 1953 4 Sheets-Sheet 3 mverrroR:

TACQUES R.v. osuA 4-D Dec. 30, 1958 J. R. v. OSWALD 5 v LATTICE-TYPE ANTIMETRIC FILTERS Filed Nov. so, 1953 4 Sheets-Sheet 4 INVENTOR',

TAcquEs R. V- OSUALD a /I. f 4 a United States Patent LATTICE-TYPE ANTIMETRIC FILTERS Jacques R." V. Oswald,

Paris, France, assignor to Cornpagnie Industrielle The present invention relates to a particular class of lattice-type reactive networks, those of which the imagennpedances W and W satisfy the relation:

R representing a given resistance.

The characteristic impedance, geometrical mean of the image-impedances, is then constant and equal to R.

The quadripoles of. which the image-impedances are bound by the Relation 1 are called antimetric.

Apart from the symmetrical four-pole networks, characterized by W =W they constitute a particularly simple class.

However, while the condition of symmetry of the lattice type networks is immediately broken up into two conditions respectively implying the equality of the two opposite groups of branches of the lattice, the condition of antimetry is of a more complex nature. The antimetric lattice type networks which are the object of the present invention are not the most common, but the simplification allowed in their structure permits the development of a theory similar to that of symmetrical networks, and the production of filters oifering new and interesting possibilities to the art.

In a lattice network, the irnpedances of the series branches can be designed by Z and Z respectively. The impedances of'the diagonalbranches can be designated by Z and Z respectively. In such case, the lattice network is antimetric if the following conditions are fulfilled:

( 1ou) lcc) 20a) 200) =R4 Where Z is the network impedance looking into the first pair of terminals if the second pair of terminals is open-circuited; Z is the impedance of the network looking in from the second pair of terminals when the first pair of terminals is open-circuited; Z is the impedance of the network looking in from the first pair of terminals when the second pair of terminals is shortcircuited; and Z is the network impedance looking in from the second pair of terminals when the first pair of terminals is short-circuited.

-Since the products (Z )(Z and (2 (Z are equal, the relation which expresses the antimetric cellcan be-written as follows:

If the impedances Z and Z are replaced by the equivalent values expressed in the terms of the impedances Z Z Z and 2,, of the lattice network, the following expression is obtained:

this condition is obviously achieved if Z Z ==Z Z =R which is the simplest case for the antimetric network. The other case which is obtained by writing the-,first of the above equation Z Z (Z +Z )+Z Z (Z +Z corresponds to Z Z =Z Z =R or (Z /Z =Z /Z This last relationship definesja balanced bridge which has an infinite attenuation at all frequencies and accordingly. is not of interest as a filter network.

Fig. 1 represents a network of this kind, comprising the adjacent branches 2 2- Z and 32 and It is at once clear that the from the terminals'll', when-the output terminals are in open circuit, andthe impedance Z seen from:the terminals 22 when the terminals 1 and 1 are shortcircuited, are" in the ratio -R- which characterizes antimetry. Thus, we have:

admittance Y seen By choosing reactances' for Z and Z a four-terminal network is obtained which has the properties of a filter.

The object of thepresent invention is precisely the determination of' the 'reactances in order to obtain a particular filter-etfect; composition of a low-pass, high filter, with infinite then the corresponding standardized impedances are as follows:

Similarly, the standardized impedances-for the network impedances Z and Z are respectively lou Z R R Utilizing these notations, the standardized image-impedances of the antimetric lattice network are given by:

210M: and 100 sent the points :Lj.

The image transfer constant is given by:

when, in order to express the attenuation function q=cth 6:

The ,basic Formulae and 6 make it possible to summarize the conditions which must be satisfied by the reactances Z1 and 2 in order to obtain a. filter of predetermined properties, thus:

A. In the pass band or bands of the filter called anticoincidence frequencies. For them q cancels out.

B. In the attenuated band or hands,

must be positive, w, being purely imaginary; hence it follows that at the frequencies for which the reactance z takes the values if the reactance Z takes the same value. These frequencies are called coincidence frequencies, for them w cancels out.

C. At the cut-off frequencies, which separate two bands in which w, is respectively real or purely imaginary, only one of the reactances takes the value if, the other taking any values differing from the preceding ones. For cutoff frequencies, 'w and q cancel out.

It can therefore be said that the frequencies for which the standard impedances'z z assume the values :1 or more briefly the points if of these reactances, play, in the theory of antimetric lattice type filters, the same part as the frequencies for which the reactances of the branches of a conventional symmetrical lattice assume the values zero or infinity, in other words the zeros and the poles of said branches.

. The preceding rules permit the specification of lattice type filters of different types (low-pass, high-pass, etc.). ;If we take a priori thefrequencies for which one of the standard reactances, z for example, takes the values if, in' other words the points if of 2 the points if of Z2 are entirely determined. Now, it is easy to show that a reactance is perfectly defined by the knowledge of its points 1-1.

Figs. 2 and 3 represent the distribution of the points if in a low-pass filter. In Fig. 2, a thin line represents the pass-band I, limited by the cut-off frequency f,,- marked by the-sign Vand a heavy line marks the attenuated band II; Circles and crosses respectively represent the zeros and poles of the standardized reactances Z1 and Z and dots surmounted by or signs repre- The cut-off frequency corresponds to a point (j) of 2;. pass band, we find a point (j) of 2 and a point of 1 (anticoincidence frequency). The attenuation function q is then zero, as also is the composite attenuation. For the frequencies f, and f (coincidence) we find respectively points and point of Z1 and 2 The image impedance W is zero (by reason of Equation 5) and the image-impedance W2 is infinity. The compositeiattenuation is infinity. It is also infinity, asis this time the image attenuation, for the frequency in, corresponding to the third intersection of he curves Z1 and Z2- 7 i i i For the frequency of the 4 Fig. 4 represents, with the same notations, the reactances of the simplest band-pass filter.

It is always possible to calculate an antimetric lattice type filter, by taking the points if of the reactances 1 and 2 However, this method is not immediately applicable, because the composite attenuation frequencies zero or infinity are not known a priori. It is first of all necessary to determine the standardized functions w and q which it is desired to use, and which are defined by the zeros of w 1 (perfect adaptation) and of -1 (double peaks of attenuation) outside the frequencies O and 00. This will often be the simplest method of determining Z and 2 Moreover, these reactances can easily be expressed as a function of W and q.

In effect we get from (5) and (6):

Let us note that:

are rational quantities and represent mutual reactances; this is always the case with \)q -1 for the antimetric four-terminal networks, but not for /w? -1; this shows the more restrictive character of the type of lattice-type four-terminal networks which are being considered. (The introduction of /w 1 and not of w1 is due to the choice of thedenomination of the branches, which is not symmetrical in relation to the two groups of terminals.) If these conditions are satisfied and if W and q are conjugated functions, that is to say, functions of which the product and the ratio are reactances, it is possible to show that z and 2 are effectively defined as reactances by the Equations 9. It is noted that Uand V are only determined to the nearest sign, but the various changes of sign only have the effect of permutating theadjacent or opposite branches of the lattice.

It is therefore sufficient to associate functions 11 and W subject to the previous conditions, and characterizing the type of filter which it is desired to obtain.

For thelow-pass filters, the choice must be made from the functions q=cth 0 which are the functions w of the g -pas fi t r The pulsations are standardized by. intro clucil lg their The impedance function w which are the attenuation ratio to. the cut-off pul atin,.and we put: functions q of the band-rejection filters, are given by the w p following expressions: (12) P:

w 6 (C The class of the functions q is defined like the degree I oimFUP Class 4) x i 'P +a d P m0+m; P [2 mt+ma 1 of (1 in p it is the numberof the simple zeros of etc. I (q?1). As an example, the specification of some filters 1S given:

Under these conditions the functions q are given by: Low pass filter 3/2vt*oc HTZIF It has the attenuati 'n function q of Class 3 and allows l (I P I a peak of double attenuation for s i V V V w o (Cl 3) VP +1 1+m P +1 Jrass P (1+ m)?P 2m+1 and a peak of simple attenuation to infinity. Taking as (Class 5) i i \/P +1 1+m1 (1+m2 P +[2 1+m1) +m2 m1+m2mP +1 P (1+m1)( z) +m1)( +m2)( +m1+m2)( 1+ 2) 1+mz)'+1 The parameters m (or m m characterize the infinite unity. the peaks of double attenuation (whichv correspond attenuation frequencies in accordance with the formula: in the ladder-type filters to a complete network, .the. points 0)? 1 of simple attenuation corresponding to the. /2 networks) I characterize it by the parameter for theattenuation. It has the image-impedances oc* and a of the table given The impedance functions w which are the attenuation above of the impedance functions of the low-pass filters. functions q=coth 0 of the high' pass filters, are defined The expressions of the characteristic functions are reby the following tables: called:

(Classl 01*) w =VP +1 U (class 3; 1 1 4. P +(1+ )2 (16) P 2 L V'Y 'w1 w/P +1( V2 2 V 2+ Pz+ 2 2 20i-1 P +a (Class 5) r 5*) wl:/P2 +1 *+i#1+uz) +u1)( +#z)] +u1) 2) etc.

I v I Carrying these values to (8) then to (9) it follows that: It is usual to represent by a) e) (Caner notations) (17) a the inverse impedances of those of the above table; it is for this reason that the impedances of saidtable are rep- 7 U= g= resented by the same letter with an asterisk signifying the T T i 1) P inversion. (l-lm)?P -|1 1-V m By means of any one of the functions of the first list 9 V:

and any of the functions w, of the second, to construct a low-pass or high pass filter. 1 a2 Similarv formulae are easily set up for theband-pass 0 2:?)

and band-rejection filters. (18) P P Thus the functions q=coth 0 of the band-pass filters, H 1 z mP (1.,a )P

which are functions w ofthe band-rejection filters, are 7 (1+ p2+ p2+" T m the form} Figured. gives the distribution of the points i j of this 0 2 2+ 1 2+ filter, Fig. 6 the structure obtained and Fig. 7 the equivalass v P +1 lent ladder-type filter. -(Class4) defined by: The simplest antimetric band-filter I (14) m Represented, with the preceding notations by P being given by: V g-b*b with: y v V V v v Anlimetrie band-filter Mb I This'filter is .definedibynmeans' of the functions w," and q 1 d- 1"). -I' given below, with the values of the angular attenuation P +1 and infinite image frequencies Q2 Q2 /(P2 -2 2+ ml; u ea (mom?) P and the calculation of the auxiliary function U and V previously defined: This filter gives simple attenuation peaks at the zero /(P +mE)(P +m) and infinite frequencies. (m -m ")P 1+mm o 2) -l-( o+ 1) o l- 2)" H P2 2 P2 2 0 1 a M t1+mm0 1+ nm P +BP +0P mn+mt mo+ml By applying the method indicated, we find: with r I M e 2 in i ooa n=% 20 a, 0., @g 1mm a: '1-m3m Q) V and f I reactances. This filter is equivalent tov the ladder-type F 4 th t f th ts d F 8 (mo+m1)( o-it)(1+ om1)( +mom=) 1g. gives e arrangemen 0 e pom an 1g. 2 2 represents the lattice-type network obtained with these B 2in ui t i+ z)( 3 1 2) Ad network, sometimes called /2 network with k con- '1 o+ 2) I Y 1 stant,. with, however, this ditference that the imageo( o F 1-l- 2)( ida a) impedances we find W and W= 1-U momma) 1 2 W g V 1+U l-l mfim m (P +c") (P wn) o( o 1+ 2)( -lt e) P*(P +c m "1)( -iu a) -l- -l' have the value, in the middle of the band: in which a, b, c, d, aregiven by:

r u" x 1 1 2)( 3)+. u(m+m;)( 1+ od- +l m (1+m m (1+ momg) W R= R i a o'l'm1)( o+ 2) mo-m w w d (l b ---''-"'-'(1+m0ml)(1+mum=) The ratio 7 M (Edi- =1 fim wfl) equal V c2 1 2 is therefore all the higher the narrower the pass-band. The I Z shows poles with the pulsations a and b, zeros with ladder-type halt network, completed by the two ratio the pulsations c, 0, do. z is an antiresonant circuit for transformers the pulsation d, v I

n \/m 1 The complete specificat on of the filter is the following:

w1'- 1 mil-172E v o 'l g i z 1 n as shown in Fig. 9, is entirely equivalent to the lattice I V R d2+ 2 z+3) type network of Fig. 8. The elements represented in 02 Figs. 8 and 9 have respectively the values: L 0 FEEL; (m m%) mg +m1m2 I C C =5, L x l @869 R o o+ 1)( e+ 2) Ree Rte-. (.0 m 7 p v =im m iwh "R1 "R=- R2= R=nm: (momr*)(1+m:mm)-

. (0l'l'. lwi- -i 7 g for Fig. 8 L (m1)(m |-m C Rm (1+m m )(1+mumz) and E oi)( o+' 2)' 2 2 t 1 1+ 2) Lo: R Fig. 10 gives the arrangement of the points if of the m-l-w-i R 'fi 3 i it ra'c'tances Z1, 2 and Fig. 11 the circuit diagram of the 1, filter.

C3= for Fig. 9 If one of the infinite attenuation frequencies ap UM- t) ;proa ches one of the. cut-ofif frequenciegthe other remain- This property of a change of impedance modulus, which ing fairly distant, this filter can be made by means of two may be remarkable for narrow-band filters, is common, quartz crystals and inductances (Fig. 12). By suitable to many lattiee-type antimetric band filters and may be 7o choice of parameters, a very wide-band filter can be of great interest in certain arrangements of selective obtained. r a amplifiers, and more generally in any circuit diagram in It is also possible to make a wide-band antimetr c filwhich it is desired to avoid introducing transformers of ter, each branch of which comprises a crystal, the filter high ratio, which are often difiicult to make (coupling 5 being of attenuation Class 4, desrgnated by the notation defect, interference capacities, etc.) 7 b*b) Figs. 13.and:14 refer to this filter.

It is interesting to observe that the symmetrical filter comprising reactances z z of the same structure, would be of Class 3 only.

These examples, of course, are not of a limitative character, the various antimetric lattice type networks, and networks with inverse standardized adjacent impedance being easily determined by means of general principles given in the description.

What I claim is:

1. In an antimetric lattice-type band pass filter having a four-terminal network provided with input and output terminals, a characteristic impedance R and an image impedance W in combination, a first pair of adjacent branches connected between one input terminal and the output terminals of the filter and respectively having impedances equal to Z and R /Z and a second pair of adjacent branches connected between another of said input terminals and the output terminals of the filter and respectively having impedances Z and R /Z said impedances of said branches having the following characteristics: (a) at the cut-off frequencies of the filter, only one of the impedances Z and Z is equal to :jR; (b) when the attenuation function of the filter is zero, each of the impedances Z and Z have the same absolute value ijR but have opposite signs; and (0) when the image impedance W is zero, the impedances Z and Z each have the same absolute value i-jR and have the same sign.

2. In an antimetric lattice-type band pass filter having a four-terminal network provided with input and output terminals, a characteristic impedance R, an input image impedance W and attenuation frequencies w and 0 in combination, a first pair of adjacent branches connected between one of said input terminals and said output terminals, said branches having impedances equal respectively to Rau /p and Rp/w where p equals the quantity jw, w equals the frequency of the voltage applied to the input terminals of the filter; and a second pair of adjacent branches connected between another of said input terminals and said output terminals of the filter and respectively having impedances Rw /p and Rp/w 3. In an antimetric lattice-type band pass filter having a four-terminal network provided with input and output 10 terminals, a characteristic impedance R, an input image impedance W and having cut-off frequencies f and f in combination, a first pair of adjacent branches connected between one of said input terminals and said out- 5 put terminals, one of said branches having an impedance Z and including a resonant circuit and a capacitor connected in parallel with an inductor, the other of said first pair of adjacent branches having an impedance equal to R /Z and a second pair of adjacent branches connected between another of said input terminals and said output terminals, one of said second pair of adjacent branches including an antiresonant circuit having an impedance Z the other of said second pair of adjacent branches having an impedance equal to R /Z said impedances of said branches having the following characteristics: (a) when the frequency of the voltage applied to the input terminals of the filter is equal to zero and infinity respectively and for at least two frequencies situated on opposite sides of the pass band, the attenuation function of the filter equals 1; (b)twhen the frequency of the applied voltage equals L the impedance Z is equal to jR; (c) when the frequency of the applied voltage is equal to f1, the impedance Z is equal to jR; and (d) for the frequency at which the impedance Z, of the antiresonant circuit equals 'R, the impedance Z equals jR; for the frequency at which the impedance Z of the antiresonant circuit equals jR, the impedance Z equals iR; said attenuation function of the filter being zero at said last two frequencies.

References Cited in the file of this patent UNITED STATES PATENTS 1,989,545 Cauer Jan. 29, 1935 2,280,282 Colchester et al Apr. 21, 1942 2,591,838 Leroy Apr. 8, 1952 lished by John Wiley & Sons, New York, in 1935, pp. 378-394. 

